A second limitation of (REF ) is that it does not account for other potential structure exhibited by the covariance or precision matrices.
This can problematic when the sample support is low (\(n\simeq p\) , or \(n<p\) ) as the sample covariance matrix is
not a reliable estimate in these setups [1]}, [2]}, [3]}.
In this context, a popular approach consists in imposing low-rank structures on the covariance matrix.
These structures arise from the assumption that the data is driven by an underlying low-dimensional linear model, i.e., \({x} = {W}{s} + {\epsilon } \) , where \({W}\)  is a rank-\(k\)  factor loading matrix, and \({s}\in \mathbb {R}^k\)  are \({\epsilon }\in \mathbb {R}^p\)  are independent random variables.
Thus the resulting covariance matrix is of the form \({\Sigma } \overset{\Delta }{=} {H} + {\Psi }\) , where \({H} = {W}\mathbb {E}[{s}{s}^\top ]{W}^\top \) 
belongs to the set of positive semi-definite matrices of rank \(k\) , denoted \(\mathcal {S}_{p,k}^+=\left\lbrace {\Sigma }\in \mathcal {S}_p ,{\Sigma } \succcurlyeq {0},{\rm rank}({\Sigma })=k\right\rbrace \) .
This model is well known in statistics, and for example, at the core of probabilistic principal component analysis, that assumes \({\Psi }\propto {I}_p\)  [4]}.
Also notice that in Laplacian-constrained models, a rank-\(k\)  precision matrix implies a \((p-k)\) -component graph [5]}.
Hence it is also interesting to leverage such spectral structures from the graph learning perspective [6]}.
